A Genetic Algorithm for Fuzzy Shortest Path in a Network with mixed fuzzy arc lengths
|
|
- Frederick Wheeler
- 5 years ago
- Views:
Transcription
1 Prodg of th 0 Itrtol Cofr o Idustrl d Oprtos Mgt Kul pur Mls Jur - 0 A Gt Algorth for Fuzz Shortst Pth Ntwork wth d fuzz r lgths z Hsszdh Irj Mhdv * Al Tjd Dprtt of Idustrl Egrg Mzdr Uvrst of S d Tholog Bol Ir Nz Mhdv-Ar Fult of Mthtl Ss Shrf Uvrst of Tholog Thr Ir * El ddrss: rjrsh@rdfflo Astrt W r ord wth th dsg of odl d lgorth for oputg shortst pth twork hvg vrous tps of fuzz r lgths Frst w dvlop w thqu for th ddto of vrous fuzz urs pth usg α -uts proposg r lst squrs odl to ot rshp futos for th osdrd ddtos Th usg rtl proposd dst futo for oprso of fuzz urs w propos w pproh to solv th fuzz ll prs shortst pth prol usg gt lgorth Epls r workd out to llustrt th ppllt of th proposd odl Kwords α ut Dst futo Shortst pth grsso Gt Algorth Itroduto Th prol of fdg shortst pth fro spfd sour od to othr od s fudtl grph thor d o tht s of otug trst [7 t G = V E grph whr V s th st of vrts ods d E s th st of dgs rs A pth tw two ods s ltrtg squ of vrts d dgs gg wth strtg d dg wth dg od Th dst ost of pth s th su of th wghts r lgths of th dgs o th pth Although ovtol grph thor th wghts of th dgs shortst pth prol SPP r ssud to prs rl urs for ost prtl ppltos howvr ths prtrs osts pts dds t t r turll prs I suh ss pproprt odg pproh justfl k us of fuzz urs d so dos th fuzz shortst pth prol FSPP ppr th ltrtur [ 6 0 Th work of Duos d Prd [ s o of th frst o ths sujt d osdrs tsos of th lssl Flod d Ford Moor Bll FMB lgorths Howvr t ws vrfd tht th lgorth would oput th shortst pth dst wthout dtfg stg pth s [ s t ws outd Kl [5 wth th fuzz dotg st d Chr [6 dfd th doto of vtl rs s g thos whos rovl fro th pth rsultd rs of th ost Blu t l [ prstd lgorth whh would fd ut vlu to lt th ur of lzd pths d th ppld odfd vrso of th k-shortst pth rsp lgorth proposd Eppst [ Followg th d of fdg fuzz st soluto Okd [ trodud th opt of th dgr of posslt of r g o shortst pth Aog th ost rt work s th o N d Pl [0 tht proposs lgorth sd o th pt d trodud Sgupt d Pl [ d whh gvs sgl fuzz shortst pth or gud for hoosg st fuzz shortst pth ordg to th dso-krs vwpot [8 Th rdr of th ppr s orgzd s follows I Sto s opts d dftos r gv Sto ps ws of oputg uts for fuzz urs W prst our fuzz su oprtor us of r lst squrs odl Sto I Sto 5 usg rtl proposd dst futo w prst gt lgorth 808
2 for fdg fuzz shortst pth d fuzz twork A pl s workd out to llustrt th ppllt of th proposd odl Sto 6 W olud Sto 7 Dftos Th α-ut d strog α-ut for fuzz ur r show d rsptvl d for [0 r dfd to : X A X A whr X s th uvrsl st Coputg α-uts for fuzz urs For th fuzz urs wth d vrtl futos th -uts r: [ [ For spf d futos th followg ss r dsussd α-uts for trpzodl fuzz urs t trpzodl fuzz ur A ut for s oputd s: 0 whr [ s th orrspodg α-ut α-uts for orl fuzz urs If s orl fuzz ur th s oputd s: 0 Fuzz pprot su oprtors Hr w us our rtl proposd pproh [ for sug vrous fuzz urs pprotl usg α- uts Th pproto s sd o fttg pproprt odl for th su usg α-uts of th ddto s th ftss dt t us dvd th α-trvl [0 to qul sutrvls lttg 0 0 = Ths w w hv st of + qudstt pots For th orl fuzz urs t s 809
3 propr to ssu α g qul to zro Thrfor ths s w osdr 0 d thus us th ozro Cut su t d th trpzodl d orl fuzz urs rsptvl Gv 0 th α-ut su of ths fuzz urs usg qutos d s otd s follows: whr [ [ Usg quto orrspodg to pots r otd for pots for th d pots for th Usg ths pots t s possl to pprot th su of th two fuzz urs A pprot rshp futo of th su s oputd fttg pproprt futo usg th α-ut pots t d d usg th pots osdr th fttg odl s Th ukow prtrs d ppr orl W rz th odl otg tht for s s th s hr for th rght hd odl d r pltl dtrd to : 5 Now slrl lt d d osdr th odl W th hv 6 7 Thus th rshp futo s dtrd to : 8 wth d s dfd 5 6 d 7 rsptvl 5 A lgorth for fuzz shortst pth twork 5 Dst tw fuzz urs 80
4 Assu tht d r two fuzz urs W ppl fuzz rkg thod for fuzz urs W hv usd ths rkg thod fftvl rt work [8t us osdr fuzz oprtos s follows: 9 It s vdt tht for o-oprl fuzz urs d th fuzz oprto rsults fuzz ur dffrt fro oth of th For pl for 509 w gt fro 9 fuzz d MV 599 whh s dffrt fro oth d To llvt ths drwk w us thod sd o th dst tw fuzz urs W us th dst futo trodud [ Th dvtgs of ths dst futo r th grlt of ts usg o vrous fuzz urs d ts rllt dstgushg uqul fuzz urs Idd th usg of th dst futo low workd out to qut pproprt for our pproh Th D p q -dst dd prtrs p d 0 q ogtv futo gv : p p p q d q d p D 0 0 p q q sup q f p 0 0 For two fuzz urs d wth orrspodg to : Dp q q If q p th th ov quto turs to: tw two fuzz urs d s 0 -uts th D p q dst s pprotl proportol p q p D To opr two fuzz r lgths d wth -uts s thr pprotos s th r supposd to rprst postv vlus w opr th wth MV 000 I ft w us forul to oput D MV d D MV d th us ths vlus for oprso of th two urs 5 Th Gt Algorth 5 Gt Algorth GA for solvg shortst pth prol wth fuzz r lgths Hr w dsr th odg sh usd th GAs Gt oprtors spf to ths odg sh r lso dfd Ths lud th tlzto rossovr d utto oprtors 5 GA Eodg Sh d Populto Itlzto How to od pth grph s rtl for dvlopg gt lgorth to ths prol Ths s ot s s s trvg sls prol to fd out turl rprstto Spl dffults rs fro pth ots vrl ur of ods d th l ur s for od grph d rdo squ of dgs usull dos ot orrspod to pth To ovro suh dffults w doptd drt pproh: od so gudg forto for ostrutg pth ut pth tslf hrooso Th pth s grtd th squtl od ppdg produr gg wth th spfd od d trtg t th spfd od A vtor p s usd to kp th tr of th ods th pth Algorth grtg tl populto Fd th vt tr Aof drtd twork G = N A dtr pop-sz d st k St l d pl= Slt r of j j A d ll t j t l l d pl=j whr j p 8
5 If j th lt =j d go to 5 Fd produd pth usg th lls th lg vtor p t k k 6 If k pop-sz th go to ls stop 5 Crossovr oprtor Crossovr os forto fro two prts suh tht th two hldr hv rs to h prt Stdrd rossovrs suh s o-pot two-pot d ufor r usd GA odls Two pths lld prts r rdol sltd fro th populto Th w slt o or two oo rs gs d rpl dffrt stos fro prts H two w hldr hroosos r grtd It s ovous tht th grtd pths r fsl 5 Mutto oprtor Th trdtol utto oprtor utts th gs vlu rdol ordg to sll prolt of utto; thus t s rl rdo wlk d dos ot gurt postv drto towrd th optl soluto Th proposd hurst utto rds ths df I ths sh th ur of pths or sltd hroosos q for th utto oprtor s dtrd usg utto oprtor s rt or proltp ; th ur of pths q s oputd to th produt of populto sz d p d q dffrt rdo urs r grtd tw d populto sz to usd s ur of sltd pths Th ur s r tw d pth lgth s sltd rdol Th opots to r- s kpt uhgd d opots r to pth lgth r rovd rplg th wth w ods otd usg Algorth 55 Evluto d slto strtg Cosdrg populto hrooso pth for h hrooso s dtrd d th pth lgth s osdrd s th hrooso vlu Th for oprso of hrooso vlus w us for fdg dsts Th w lgorth for fdg shortst pth follows hr Algorth fdg shortst pth Stp Fd th possl pths fro sour vrt s to dstto vrt d fro og produd populto h rptto d oput th orrspodg pth lgths = for th possl pths Stp Fd th fuzz shortst lgth th followg stps: Stp - St Stp - for do Stp - Clult MV M Eq 9 Stp - Fd th dst D p q of MV fro d usg Eq : D D p q MV D D p q MV rg D D 6 Nurl Illustrtos Epl :W osdr lrgr twork hvg d r lgths oto of orl d trpzodl fuzz urs d us our gt lgorth to fd th shortst pths Th r lgths r spfd Tl Tl Th r lgths r fuzz ur r fuzz ur r fuzz ur r fuzz ur r fuzz ur
6 Usg th dst futo D p q od s dtrd to : Fgur shows th ovrg urv for Epl for q=/ d p= th shortst pth fro th sour od to th dstto Shortst pth lgths Fgur Covrg urv for Epl Itrtos B ddto of vrous fuzz urs o th orrspodg pth th rshp futo s otd -7 s follows: μ = < < < 65 > 65 7 Colusos W osdrd th prol of fdg th ll-prs fuzz shortst pths wth th lgths of r gv fuzz urs Frst w proposd ordr rlto tw fuzz urs Th w dvlopd fuzz rkg thod to vod grtg th st of o-dotd pths or Prto optl pths us th ur of o-dotd pths drvd fro lrg twork too urous d t ould dffult for dso kr to hoos prfrl pth Th w vstgtd th posslt of usg gt lgorth to solv fuzz shortst pth prols Th proposd pproh ws llustrtd o rdol grtd prol hvg ods d 0 rs frs [ M Blu B Bush J Puktt Ufd pproh to fuzz grph prols Fuzz Sts d Ssts [ T-N Chug J-Y Kug Th fuzz shortst pth lgth d th orrspodg shortst pth twork Coput Opr s [ D Duos H Prd Fuzz Sts d Ssts: Thor d Appltos Ad Prss Nw York 980 [ D Eppst Fdg th k-shortst pths : Pro IEEE Sp o Foudtos of Coputr S [5 CM Kl Fuzz shortst pths Fuzz Sts d Ssts [6 C MS Chr Th fuzz shortst pth prol d ts ost vtl rs Fuzz Sts d Ssts [7 MT Tkhsh Cotruçõs o studo d grfos fuzz: Tor lgortos PhD Thss Fuldd d Eghr Elétr d Coputção UNICAMP 00 [8 I Mhdv Nourfr A Hdrzd N Mhdv Ar A d progrg pproh for fdg shortst hs fuzz twork Appld Soft Coputg [9 JA Moro JM Moro J Vrdg Fuzz loto prols o tworks Fuzz Sts d Ssts [0 SMA N M Pl Shortst pth prol o twork wth prs dg wght Fuzz Opt Ds Mkg [ S Okd Fuzz shortst pth prols orportg trtvt og pths Fuzz Sts d Ssts [ B Sdghpour Gldh D G Dst-Dpq t l Cofft d Corrélto tr du Vrls Alétors flous Ats d FA Mos-Blgu 00 [ A Sgupt TK Pl O oprg trvl urs Europ J Opr s [ A Tjd I Mhdv N Mhdv-Ar B Sdghpour-Gldh Coputg fuzz shortst pth twork wth d fuzz r lgths usg α-uts Coputrs d Mthts wth Appltos prss 00 8
minimize c'x subject to subject to subject to
z ' sut to ' M ' M N uostrd N z ' sut to ' z ' sut to ' sl vrls vtor of : vrls surplus vtor of : uostrd s s s s s s z sut to whr : ut ost of :out of : out of ( ' gr of h food ( utrt : rqurt for h utrt
More informationExtension Formulas of Lauricella s Functions by Applications of Dixon s Summation Theorem
Avll t http:pvu.u Appl. Appl. Mth. ISSN: 9-9466 Vol. 0 Issu Dr 05 pp. 007-08 Appltos Appl Mthts: A Itrtol Jourl AAM Etso oruls of Lurll s utos Appltos of Do s Suto Thor Ah Al Atsh Dprtt of Mthts A Uvrst
More informationHandout 11. Energy Bands in Graphene: Tight Binding and the Nearly Free Electron Approach
Hdout rg ds Grh: Tght dg d th Nrl Fr ltro roh I ths ltur ou wll lr: rg Th tght bdg thod (otd ) Th -bds grh FZ C 407 Srg 009 Frh R Corll Uvrst Grh d Crbo Notubs: ss Grh s two dsol sgl to lr o rbo tos rrgd
More informationCBSE SAMPLE PAPER SOLUTIONS CLASS-XII MATHS SET-2 CBSE , ˆj. cos. SECTION A 1. Given that a 2iˆ ˆj. We need to find
BSE SMLE ER SOLUTONS LSS-X MTHS SET- BSE SETON Gv tht d W d to fd 7 7 Hc, 7 7 7 Lt, W ow tht Thus, osd th vcto quto of th pl z - + z = - + z = Thus th ts quto of th pl s - + z = Lt d th dstc tw th pot,,
More informationCBSE , ˆj. cos CBSE_2015_SET-1. SECTION A 1. Given that a 2iˆ ˆj. We need to find. 3. Consider the vector equation of the plane.
CBSE CBSE SET- SECTION. Gv tht d W d to fd 7 7 Hc, 7 7 7. Lt,. W ow tht.. Thus,. Cosd th vcto quto of th pl.. z. - + z = - + z = Thus th Cts quto of th pl s - + z = Lt d th dstc tw th pot,, - to th pl.
More informationSection 5.1/5.2: Areas and Distances the Definite Integral
Scto./.: Ars d Dstcs th Dt Itgrl Sgm Notto Prctc HW rom Stwrt Ttook ot to hd p. #,, 9 p. 6 #,, 9- odd, - odd Th sum o trms,,, s wrtt s, whr th d o summto Empl : Fd th sum. Soluto: Th Dt Itgrl Suppos w
More informationFace Detection and Recognition. Linear Algebra and Face Recognition. Face Recognition. Face Recognition. Dimension reduction
F Dtto Roto Lr Alr F Roto C Y I Ursty O solto: tto o l trs s s ys os ot. Dlt to t to ltpl ws. F Roto Aotr ppro: ort y rry s tor o so E.. 56 56 > pot 6556- stol sp A st o s t ps to ollto o pots ts sp. F
More informationTHE TRANSMUTED GENERALIZED PARETO DISTRIBUTION. STATISTICAL INFERENCE AND SIMULATION RESULTS
Mr l Btr vl Admy Stf Bullt Volum VIII 5 Issu Pulshd y Mr l Btr vl Admy Prss Costt Rom // Th jourl s dd : PROQUST STh Jourls PROQUST grg Jourls PROQUST Illustrt: Thology PROQUST Thology Jourls PROQUST Mltry
More informationOn the Existence and uniqueness for solution of system Fractional Differential Equations
OSR Jourl o Mhms OSR-JM SSN: 78-578. Volum 4 ssu 3 Nov. - D. PP -5 www.osrjourls.org O h Es d uquss or soluo o ssm rol Drl Equos Mh Ad Al-Wh Dprm o Appld S Uvrs o holog Bghdd- rq Asr: hs ppr w d horm o
More informationERDOS-SMARANDACHE NUMBERS. Sabin Tabirca* Tatiana Tabirca**
ERDO-MARANDACHE NUMBER b Tbrc* Tt Tbrc** *Trslv Uvrsty of Brsov, Computr cc Dprtmt **Uvrsty of Mchstr, Computr cc Dprtmt Th strtg pot of ths rtcl s rprstd by rct work of Fch []. Bsd o two symptotc rsults
More informationPhoton-phonon interaction in photonic crystals
IOP Cofr Srs: Mtrls S d Egrg Photo-phoo trto photo rystls To t ths rtl: T Ut IOP Cof. Sr.: Mtr. S. Eg. 3 Vw th rtl ol for updts d hmts. Rltd ott - study o optl proprts of photo rystls osstg of hollow rods
More informationOptimized Robotic Assembly Sequence using ACO
S.Shr Lhlr I Prvt Lt., Th (Wst), Mu, 600604, I surjtt@gl.o Opt Root Assl Squ usg ACO B.B.Bswl Dpt. of Mh. Egg. NIT Rourl Orss I hut.swl@gl.o P.Dsh Dpt. of Mh. Egg. IACR, Rg Orss I prswr_sh@hoo.o B.B.Chouhur
More informationFinite Element Approach to Electric Field Distribution Resulting from Phase-sequence Orientation of a Double- Circuit High Voltage Transmission Line
Prodgs of th 8th WSEAS Itrtol Cofr o ELECTRIC POWER SYSTEMS HIGH VOLTAGES ELECTRIC MACHIES (POWER '8) Ft Elt Approh to Eltr Fld Dstruto Rsultg fro Phs-squ Ortto of Doul- Crut Hgh Voltg Trssso L A. ISARAMOGKOLRAK
More informationSpectral Characteristics of Digitally Modulated Signals
Strl Chrtrt of Dgtlly odultd Sgl 6:33:56 Wrl Couto holog Srg 5 Ltur7&8 Drtt of Eltrl Egrg Rutgr Uvrty Ptwy J 89 ught y Dr. ry dy ry@wl.rutgr.du Doutd y Bozh Yu ozh@d.rutgr.du trt: h ltur frt trodu th tdrd
More informationBEM with Linear Boundary Elements for Solving the Problem of the 3D Compressible Fluid Flow around Obstacles
EM wth L ou Elts o olvg th Pol o th D opssl Flu Flow ou Ostls Lut Gu o Vlsu stt hs pp psts soluto o th sgul ou tgl quto o th D opssl lu low ou ostl whh uss sopt l ou lts o Lgg tp. h sgul ou tgl quto oult
More informationInner Product Spaces INNER PRODUCTS
MA4Hcdoc Ir Product Spcs INNER PRODCS Dto A r product o vctor spc V s ucto tht ssgs ubr spc V such wy tht th ollowg xos holds: P : w s rl ubr P : P : P 4 : P 5 : v, w = w, v v + w, u = u + w, u rv, w =
More informationLinear Prediction Analysis of Speech Sounds
Lr Prdcto Alyss of Sch Souds Brl Ch 4 frcs: X Hug t l So Lgug Procssg Chtrs 5 6 J Dllr t l Dscrt-T Procssg of Sch Sgls Chtrs 4-6 3 J W Pco Sgl odlg tchqus sch rcogto rocdgs of th I Stbr 993 5-47 Lr Prdctv
More informationSpecial Curves of 4D Galilean Space
Irol Jourl of Mhml Egrg d S ISSN : 77-698 Volum Issu Mrh hp://www.jms.om/ hps://ss.googl.om/s/jmsjourl/ Spl Curvs of D ll Sp Mhm Bkş Mhmu Ergü Alpr Osm Öğrmş Fır Uvrsy Fuly of S Dprm of Mhms 9 Elzığ Türky
More informationHandout on. Crystal Symmetries and Energy Bands
dou o Csl s d g Bds I hs lu ou wll l: Th loshp bw ss d g bds h bs of sp-ob ouplg Th loshp bw ss d g bds h ps of sp-ob ouplg C 7 pg 9 Fh Coll Uvs d g Bds gll hs oh Th sl pol ss ddo o h l slo s: Fo pl h
More informationLet's revisit conditional probability, where the event M is expressed in terms of the random variable. P Ax x x = =
L's rvs codol rol whr h v M s rssd rs o h rdo vrl. L { M } rrr v such h { M } Assu. { } { A M} { A { } } M < { } { } A u { } { } { A} { A} ( A) ( A) { A} A A { A } hs llows us o cosdr h cs wh M { } [ (
More informationMajor: All Engineering Majors. Authors: Autar Kaw, Luke Snyder
Nolr Rgrsso Mjor: All Egrg Mjors Auhors: Aur Kw, Luk Sydr hp://urclhodsgusfdu Trsforg Nurcl Mhods Educo for STEM Udrgrdus 3/9/5 hp://urclhodsgusfdu Nolr Rgrsso hp://urclhodsgusfdu Nolr Rgrsso So populr
More informationIII Z-Plane Analysis
III Z-Pl Aly opc to covrd. Itroducto. Ipul plg d dt hold 3. Otg th Z trfor y covoluto 4. Sgl rcotructo 5. h pul trfr fucto 6. Dgtl cotrollr d fltr III. Itroducto h dvtg of th trfor thod tht t l th gr to
More informationOn Matrices associated with L-Fuzzy Graphs
lol Jorl of Pr d Appld Mthmts ISSN 973-768 olm 3 Nmr 6 7 pp 799-8 Rsrh Id Pltos http://wwwrpltoom O Mtrs ssotd wth -Fzzy rphs Prmd Rmhdr P Dprtmt of Mthmts St Pl s Collg Klmssry Koh-683 53 Krl Id K Thoms
More informationStudy on Non-linear Responses of Eccentric Structure
Th 4 h World ofr o Erh Egrg or -7 8 Bg h Sd o No-lr Rpo of Er Srr Hdz WATANABE oh USUNI Ar TASAI 3 Grd Sd Dpr of Arhr ooh Nol Uvr ooh Jp Ao Profor Dpr of Arhr ooh Nol Uvr ooh Jp ABSTRAT : 3 Profor Dpr
More informationLinear Prediction Analysis of
Lr Prdcto Alyss of Sch Souds Brl Ch Drtt of Coutr Scc & Iforto grg Ntol Tw Norl Uvrsty frcs: X Hug t l So Lgug g Procssg Chtrs 5 6 J Dllr t l Dscrt-T Procssg of Sch Sgls Chtrs 4-6 3 J W Pco Sgl odlg tchqus
More informationAnalele Universităţii din Oradea, Fascicula: Protecţia Mediului, Vol. XIII, 2008
Alele Uverstăţ d Orde Fsul: Proteţ Medulu Vol. XIII 00 THEORETICAL AND COMPARATIVE STUDY REGARDING THE MECHANICS DISPLASCEMENTS UNDER THE STATIC LOADINGS FOR THE SQUARE PLATE MADE BY WOOD REFUSE AND MASSIF
More informationKummer Beta -Weibull Geometric Distribution. A New Generalization of Beta -Weibull Geometric Distribution
ttol Jol of Ss: Bs Al Rsh JSBAR SSN 37-453 Pt & Ol htt://gss.og/.h?joljolofbsaal ---------------------------------------------------------------------------------------------------------------------------
More informationThe linear system. The problem: solve
The ler syste The prole: solve Suppose A s vertle, the there ests uue soluto How to effetly opute the soluto uerlly??? A A A evew of dret ethods Guss elto wth pvotg Meory ost: O^ Coputtol ost: O^ C oly
More informationLECTURE 6 TRANSFORMATION OF RANDOM VARIABLES
LECTURE 6 TRANSFORMATION OF RANDOM VARIABLES TRANSFORMATION OF FUNCTION OF A RANDOM VARIABLE UNIVARIATE TRANSFORMATIONS TRANSFORMATION OF RANDOM VARIABLES If s a rv wth cdf F th Y=g s also a rv. If w wrt
More informationCOMP108 Algorithmic Foundations
Grdy mthods Prudn Wong http://www.s.liv..uk/~pwong/thing/omp108/01617 Coin Chng Prolm Suppos w hv 3 typs of oins 10p 0p 50p Minimum numr of oins to mk 0.8, 1.0, 1.? Grdy mthod Lrning outoms Undrstnd wht
More informationCOMPLEX NUMBERS AND ELEMENTARY FUNCTIONS OF COMPLEX VARIABLES
COMPLEX NUMBERS AND ELEMENTARY FUNCTIONS OF COMPLEX VARIABLES DEFINITION OF A COMPLEX NUMBER: A umbr of th form, whr = (, ad & ar ral umbrs s calld a compl umbr Th ral umbr, s calld ral part of whl s calld
More informationCURVE FITTING LEAST SQUARES METHOD
Nuercl Alss for Egeers Ger Jord Uverst CURVE FITTING Although, the for of fucto represetg phscl sste s kow, the fucto tself ot be kow. Therefore, t s frequetl desred to ft curve to set of dt pots the ssued
More informationHIGHER ORDER DIFFERENTIAL EQUATIONS
Prof Enriqu Mtus Nivs PhD in Mthmtis Edution IGER ORDER DIFFERENTIAL EQUATIONS omognous linr qutions with onstnt offiints of ordr two highr Appl rdution mthod to dtrmin solution of th nonhomognous qution
More informationR. 7.5 E. R. 8 E. ! ( y R. S a Clackamas County. . Sa. Zi gzag R. S almon R. U.S. Forest Service 63. acka m a. Wasco. County. Jefferson. County.
T 2 N r o T 1 N T 1 T 2 B d r C r lo B L t t dr vr Wh t E Wh o c r C l T 27 ch C r L r c t f C r T 4 Z z t h E T 3 H o od y d Clc l 36 C 4 N t T 3 E C r l E N 17 E H o od u u ll B ull u T 1 B 3 vr M H
More informationSAMPLE CSc 340 EXAM QUESTIONS WITH SOLUTIONS: part 2
AMPLE C EXAM UETION WITH OLUTION: prt. It n sown tt l / wr.7888l. I Φ nots orul or pprotng t vlu o tn t n sown tt t trunton rror o ts pproton s o t or or so onstnts ; tt s Not tt / L Φ L.. Φ.. /. /.. Φ..787.
More informationOn Hamiltonian Tetrahedralizations Of Convex Polyhedra
O Ht Ttrrzts O Cvx Pyr Frs C 1 Q-Hu D 2 C A W 3 1 Dprtt Cputr S T Uvrsty H K, H K, C. E: @s.u. 2 R & TV Trsss Ctr, Hu, C. E: q@163.t 3 Dprtt Cputr S, Mr Uvrsty Nwu St. J s, Nwu, C A1B 35. E: w@r.s.u. Astrt
More informationChapter 16. 1) is a particular point on the graph of the function. 1. y, where x y 1
Prctic qustions W now tht th prmtr p is dirctl rltd to th mplitud; thrfor, w cn find tht p. cos d [ sin ] sin sin Not: Evn though ou might not now how to find th prmtr in prt, it is lws dvisl to procd
More informationLinear Algebra Existence of the determinant. Expansion according to a row.
Lir Algbr 2270 1 Existc of th dtrmit. Expsio ccordig to row. W dfi th dtrmit for 1 1 mtrics s dt([]) = (1) It is sy chck tht it stisfis D1)-D3). For y othr w dfi th dtrmit s follows. Assumig th dtrmit
More informationDEVELOPING COMPUTER PROGRAM FOR COMPUTING EIGENPAIRS OF 2 2 MATRICES AND 3 3 UPPER TRIANGULAR MATRICES USING THE SIMPLE ALGORITHM
Fr Est Journl o Mthtil Sins (FJMS) Volu 6 Nur Pgs 8- Pulish Onlin: Sptr This ppr is vill onlin t http://pphjo/journls/jsht Pushp Pulishing Hous DEVELOPING COMPUTER PROGRAM FOR COMPUTING EIGENPAIRS OF MATRICES
More informationHandout 7. Properties of Bloch States and Electron Statistics in Energy Bands
Hdout 7 Popts of Bloch Stts d Elcto Sttstcs Eg Bds I ths lctu ou wll l: Popts of Bloch fuctos Podc boud codtos fo Bloch fuctos Dst of stts -spc Elcto occupto sttstcs g bds ECE 407 Spg 009 Fh R Coll Uvst
More informationSYSTEMS OF LINEAR EQUATIONS
SYSES OF INER EQUIONS Itroducto Emto thods Dcomposto thods tr Ivrs d Dtrmt Errors, Rsdus d Codto Numr Itrto thods Icompt d Rdudt Systms Chptr Systms of r Equtos /. Itroducto h systm of r qutos s formd
More informationPlanar convex hulls (I)
Covx Hu Covxty Gv st P o ots 2D, tr ovx u s t sst ovx oyo tt ots ots o P A oyo P s ovx or y, P, t st s try P. Pr ovx us (I) Coutto Gotry [s 3250] Lur To Bowo Co ovx o-ovx 1 2 3 Covx Hu Covx Hu Covx Hu
More informationA Review of Dynamic Models Used in Simulation of Gear Transmissions
ANALELE UNIVERSITĂłII ETIMIE MURGU REŞIłA ANUL XXI NR. ISSN 5-797 Zol-Ios Ko Io-ol Mulu A Rvw o ls Us Sulo o G Tsssos Th vsgo o lv s lu gg g olg l us o sov sg u o pps g svl s oug o h ps. Th pupos o h ols
More informationNumerical Method: Finite difference scheme
Numrcal Mthod: Ft dffrc schm Taylor s srs f(x 3 f(x f '(x f ''(x f '''(x...(1! 3! f(x 3 f(x f '(x f ''(x f '''(x...(! 3! whr > 0 from (1, f(x f(x f '(x R Droppg R, f(x f(x f '(x Forward dffrcg O ( x from
More informationUnbalanced Panel Data Models
Ubalacd Pal Data odls Chaptr 9 from Baltag: Ecoomtrc Aalyss of Pal Data 5 by Adrás alascs 4448 troducto balacd or complt pals: a pal data st whr data/obsrvatos ar avalabl for all crosssctoal uts th tr
More informationH NT Z N RT L 0 4 n f lt r h v d lt n r n, h p l," "Fl d nd fl d " ( n l d n l tr l t nt r t t n t nt t nt n fr n nl, th t l n r tr t nt. r d n f d rd n t th nd r nt r d t n th t th n r lth h v b n f
More informationHaving a glimpse of some of the possibilities for solutions of linear systems, we move to methods of finding these solutions. The basic idea we shall
Hvn lps o so o t posslts or solutons o lnr systs, w ov to tos o nn ts solutons. T s w sll us s to try to sply t syst y lntn so o t vrls n so ts qutons. Tus, w rr to t to s lnton. T prry oprton nvolv s
More information1. Stefan-Boltzmann law states that the power emitted per unit area of the surface of a black
Stf-Boltzm lw stts tht th powr mttd pr ut r of th surfc of blck body s proportol to th fourth powr of th bsolut tmprtur: 4 S T whr T s th bsolut tmprtur d th Stf-Boltzm costt= 5 4 k B 3 5c h ( Clcult 5
More informationPriority Search Trees - Part I
.S. 252 Pro. Rorto Taassa oputatoal otry S., 1992 1993 Ltur 9 at: ar 8, 1993 Sr: a Q ol aro Prorty Sar Trs - Part 1 trouto t last ltur, w loo at trval trs. or trval pot losur prols, ty us lar spa a optal
More informationDepartment of Mathematics and Statistics Indian Institute of Technology Kanpur MSO202A/MSO202 Assignment 3 Solutions Introduction To Complex Analysis
Dpartmt of Mathmatcs ad Statstcs Ida Isttut of Tchology Kapur MSOA/MSO Assgmt 3 Solutos Itroducto To omplx Aalyss Th problms markd (T) d a xplct dscusso th tutoral class. Othr problms ar for hacd practc..
More informationP a g e 5 1 of R e p o r t P B 4 / 0 9
P a g e 5 1 of R e p o r t P B 4 / 0 9 J A R T a l s o c o n c l u d e d t h a t a l t h o u g h t h e i n t e n t o f N e l s o n s r e h a b i l i t a t i o n p l a n i s t o e n h a n c e c o n n e
More informationFormal Concept Analysis
Forml Conpt Anlysis Conpt intnts s losd sts Closur Systms nd Implitions 4 Closur Systms 0.06.005 Nxt-Closur ws dvlopd y B. Gntr (984). Lt M = {,..., n}. A M is ltilly smllr thn B M, if B A if th smllst
More informationMath 61 : Discrete Structures Final Exam Instructor: Ciprian Manolescu. You have 180 minutes.
Nm: UCA ID Numr: Stion lttr: th 61 : Disrt Struturs Finl Exm Instrutor: Ciprin nolsu You hv 180 minuts. No ooks, nots or lultors r llow. Do not us your own srth ppr. 1. (2 points h) Tru/Fls: Cirl th right
More informationGUC (Dr. Hany Hammad) 9/28/2016
U (r. Hny Hd) 9/8/06 ctur # 3 ignl flow grphs (cont.): ignl-flow grph rprsnttion of : ssiv sgl-port dvic. owr g qutions rnsducr powr g. Oprtg powr g. vill powr g. ppliction to Ntwork nlyzr lirtion. Nois
More informationCHAPTER 4. FREQUENCY ESTIMATION AND TRACKING
CHPTER 4. FREQUENCY ESTITION ND TRCKING 4.. Itroducto Estmtg mult-frquc susodl sgls burd os hs b th focus of rsrch for qut som tm [68] [58] [46] [64]. ost of th publshd rsrch usd costrd ft mpuls rspos
More informationADORO TE DEVOTE (Godhead Here in Hiding) te, stus bat mas, la te. in so non mor Je nunc. la in. tis. ne, su a. tum. tas: tur: tas: or: ni, ne, o:
R TE EVTE (dhd H Hdg) L / Mld Kbrd gú s v l m sl c m qu gs v nns V n P P rs l mul m d lud 7 súb Fí cón ví f f dó, cru gs,, j l f c r s m l qum t pr qud ct, us: ns,,,, cs, cut r l sns m / m fí hó sn sí
More informationPaths. Connectivity. Euler and Hamilton Paths. Planar graphs.
Pths.. Eulr n Hmilton Pths.. Pth D. A pth rom s to t is squn o gs {x 0, x 1 }, {x 1, x 2 },... {x n 1, x n }, whr x 0 = s, n x n = t. D. Th lngth o pth is th numr o gs in it. {, } {, } {, } {, } {, } {,
More informationBayesian belief networks: Inference
C 740 Knowd rprntton ctur 0 n f ntwork: nfrnc o ukrcht o@c.ptt.du 539 nnott qur C 750 chn rnn n f ntwork. 1. Drctd ccc rph Nod rndo vr nk n nk ncod ndpndnc. urr rthquk r ohnc rc C 750 chn rnn n f ntwork.
More information(a) v 1. v a. v i. v s. (b)
Outlin RETIMING Struturl optimiztion mthods. Gionni D Mihli Stnford Unirsity Rtiming. { Modling. { Rtiming for minimum dly. { Rtiming for minimum r. Synhronous Logi Ntwork Synhronous Logi Ntwork Synhronous
More information(2) If we multiplied a row of B by λ, then the value is also multiplied by λ(here lambda could be 0). namely
. DETERMINANT.. Dtrminnt. Introution:I you think row vtor o mtrix s oorint o vtors in sp, thn th gomtri mning o th rnk o th mtrix is th imnsion o th prlllppi spnn y thm. But w r not only r out th imnsion,
More informationOutline. Outline. Outline. Questions 2010/9/30. Introduction The Multivariate Normal Density and Its Properties
9 Multvrt orml Dstruto Shyh-Kg Jg Drtmt of Eltrl Egrg Grdut Isttut of Commuto Grdut Isttut of tworkg d Multmd Outl Itroduto Th Multvrt orml Dsty d Its Prorts Smlg from Multvrt orml Dstruto d Mmum Lklhood
More informationQuantum Circuits. School on Quantum Day 1, Lesson 5 16:00-17:00, March 22, 2005 Eisuke Abe
Qutum Crcuts School o Qutum Computg @Ygm D, Lsso 5 6:-7:, Mrch, 5 Esuk Ab Dprtmt of Appl Phscs Phsco-Iformtcs, CEST-JST, Ko vrst Outl Bloch sphr rprstto otto gts vrslt proof A rbtrr cotroll- gt c b mplmt
More informationCounting the compositions of a positive integer n using Generating Functions Start with, 1. x = 3 ), the number of compositions of 4.
Coutg th compostos of a postv tgr usg Gratg Fuctos Start wth,... - Whr, for ampl, th co-ff of s, for o summad composto of aml,. To obta umbr of compostos of, w d th co-ff of (...) ( ) ( ) Hr for stac w
More information2. Elementary Linear Algebra Problems
. Eleety e lge Pole. BS: B e lge Suoute (Pog pge wth PCK) Su of veto opoet:. Coputto y f- poe: () () () (3) N 3 4 5 3 6 4 7 8 Full y tee Depth te tep log()n Veto updte the f- poe wth N : ) ( ) ( ) ( )
More informationDual-Matrix Approach for Solving the Transportation Problem
Itertol Jourl of Mthets Tres Tehology- Volue Nuer Jue 05 ul-mtr Aroh for Solvg the Trsortto Prole Vy Shr r Chr Bhus Shr ertet of Mthets, BBM College r, Jeh, (MU), INIA E-Prl, SS College Jeh, (MU), INIA
More informationME 501A Seminar in Engineering Analysis Page 1
St Ssts o Ordar Drtal Equatos Novbr 7 St Ssts o Ordar Drtal Equatos Larr Cartto Mcacal Er 5A Sar Er Aalss Novbr 7 Outl Mr Rsults Rvw last class Stablt o urcal solutos Stp sz varato or rror cotrol Multstp
More informationBayesian Estimation of the parameters of the Weibull-Weibull Length-Biased mixture distributions using time censored data
Bys Eso of h s of h Wull-Wull gh-bs xu suos usg so S. A. Sh N Bouss I.S.S. Co Uvsy I.N.P.S. Algs Uvsy shsh@yhoo.o ou005@yhoo.o As I hs h s of h Wull-Wull lgh s xu suos s usg h Gs slg hqu u y I sog sh.
More informationRectangular Waveguides
Rtgulr Wvguids Wvguids tt://www.tllguid.o/wvguidlirit.tl Uss To rdu ttutio loss ig rquis ig owr C ort ol ov rti rquis Ats s ig-ss iltr Norll irulr or rtgulr W will ssu losslss rtgulr tt://www..surr..u/prsol/d.jris/wguid.tl
More informationThe Z transform techniques
h Z trnfor tchniqu h Z trnfor h th rol in dicrt yt tht th Lplc trnfor h in nlyi of continuou yt. h Z trnfor i th principl nlyticl tool for ingl-loop dicrt-ti yt. h Z trnfor h Z trnfor i to dicrt-ti yt
More informationNumerical Analysis Topic 4: Least Squares Curve Fitting
Numerl Alss Top 4: Lest Squres Curve Fttg Red Chpter 7 of the tetook Alss_Numerk Motvto Gve set of epermetl dt: 3 5. 5.9 6.3 The reltoshp etwee d m ot e ler. Fd futo f tht est ft the dt 3 Alss_Numerk Motvto
More informationCHAPTER 7. X and 2 = X
CHATR 7 Sco 7-7-. d r usd smors o. Th vrcs r d ; comr h S vrc hs cs / / S S Θ Θ Sc oh smors r usd mo o h vrcs would coclud h s h r smor wh h smllr vrc. 7-. [ ] Θ 7 7 7 7 7 7 [ ] Θ ] [ 7 6 Boh d r usd sms
More information3.4 Properties of the Stress Tensor
cto.4.4 Proprts of th trss sor.4. trss rasformato Lt th compots of th Cauchy strss tsor a coordat systm wth bas vctors b. h compots a scod coordat systm wth bas vctors j,, ar gv by th tsor trasformato
More information6.6 Moments and Centers of Mass
th 8 www.tetodre.co 6.6 oets d Ceters of ss Our ojectve here s to fd the pot P o whch th plte of gve shpe lces horzotll. Ths pot s clled the ceter of ss ( or ceter of grvt ) of the plte.. We frst cosder
More informationSheet Title: Building Renderings M. AS SHOWN Status: A.R.H.P.B. SUBMITTAL August 9, :07 pm
1 2 3 4 5 6 7 8 9 1 11 12 13 14 15 16 17 18 19 orthstar expressly reserves its common law copyright and other property rights for all ideas, provisions and plans represented or indicated by these drawings,
More informationECE COMBINATIONAL BUILDING BLOCKS - INVEST 13 DECODERS AND ENCODERS
C 24 - COMBINATIONAL BUILDING BLOCKS - INVST 3 DCODS AND NCODS FALL 23 AP FLZ To o "wll" on this invstition you must not only t th riht nswrs ut must lso o nt, omplt n onis writups tht mk ovious wht h
More informationA general N-dimensional vector consists of N values. They can be arranged as a column or a row and can be real or complex.
Lnr lgr Vctors gnrl -dmnsonl ctor conssts of lus h cn rrngd s column or row nd cn rl or compl Rcll -dmnsonl ctor cn rprsnt poston, loct, or cclrton Lt & k,, unt ctors long,, & rspctl nd lt k h th componnts
More information4 8 N v btr 20, 20 th r l f ff nt f l t. r t pl n f r th n tr t n f h h v lr d b n r d t, rd n t h h th t b t f l rd n t f th rld ll b n tr t d n R th
n r t d n 20 2 :24 T P bl D n, l d t z d http:.h th tr t. r pd l 4 8 N v btr 20, 20 th r l f ff nt f l t. r t pl n f r th n tr t n f h h v lr d b n r d t, rd n t h h th t b t f l rd n t f th rld ll b n
More informationn r t d n :4 T P bl D n, l d t z d th tr t. r pd l
n r t d n 20 20 :4 T P bl D n, l d t z d http:.h th tr t. r pd l 2 0 x pt n f t v t, f f d, b th n nd th P r n h h, th r h v n t b n p d f r nt r. Th t v v d pr n, h v r, p n th pl v t r, d b p t r b R
More informationLuiz Leal Oak Ridge National Laboratory. of Massachusetts Institute. of Technology (MIT)
LzLl OkRdgNlLby LsPsdhNl Egg Dp f h MsshssIs f Thlgy(MIT) Csy f Lz Ll, Ok Rdg Nl Lby. Usd wh pss. NI T Idpd Tsp Eq f Φ(E,,Ωˆ ) Ωˆ. Φ + Σ Φ = dωˆ ' de'σ s (E' E, Ωˆ ' Ω)Φ(E', ',Ωˆ ) + S 4 π 0 Σ Msplsss
More information1 Introduction to Modulo 7 Arithmetic
1 Introution to Moulo 7 Arithmti Bor w try our hn t solvin som hr Moulr KnKns, lt s tk los look t on moulr rithmti, mo 7 rithmti. You ll s in this sminr tht rithmti moulo prim is quit irnt rom th ons w
More informationComparisons of the Variance of Predictors with PPS sampling (update of c04ed26.doc) Ed Stanek
Coparo o th Varac o Prdctor wth PPS aplg (updat o c04d6doc Ed Sta troducto W copar prdctor o a PSU a or total bad o PPS aplg Th tratgy to ollow that o Sta ad Sgr (JASA, 004 whr w xpr th prdctor a a lar
More informationPreview. Graph. Graph. Graph. Graph Representation. Graph Representation 12/3/2018. Graph Graph Representation Graph Search Algorithms
/3/0 Prvw Grph Grph Rprsntton Grph Srch Algorthms Brdth Frst Srch Corrctnss of BFS Dpth Frst Srch Mnmum Spnnng Tr Kruskl s lgorthm Grph Drctd grph (or dgrph) G = (V, E) V: St of vrt (nod) E: St of dgs
More informationVr Vr
F rt l Pr nt t r : xt rn l ppl t n : Pr nt rv nd PD RDT V t : t t : p bl ( ll R lt: 00.00 L n : n L t pd t : 0 6 20 8 :06: 6 pt (p bl Vr.2 8.0 20 8.0. 6 TH N PD PPL T N N RL http : h b. x v t h. p V l
More informationFactors Success op Ten Critical T the exactly what wonder may you referenced, being questions different the all With success critical ten top the of l
Fr Su p T rl T xl r rr, bg r ll Wh u rl p l Fllg ll r lkg plr plr rl r kg: 1 k r r u v P 2 u l r P 3 ) r rl k 4 k rprl 5 6 k prbl lvg hkg rl 7 lxbl F 8 l S v 9 p rh L 0 1 k r T h r S pbl r u rl bv p p
More informationQuantum Harmonic Oscillator
Quu roc Oscllor Quu roc Oscllor 6 Quu Mccs Prof. Y. F. C Quu roc Oscllor Quu roc Oscllor D S..O.:lr rsorg forc F k, k s forc cos & prbolc pol. V k A prcl oscllg roc pol roc pol s u po of sbly sys 6 Quu
More informationFOURIER SERIES. Series expansions are a ubiquitous tool of science and engineering. The kinds of
Do Bgyoko () FOURIER SERIES I. INTRODUCTION Srs psos r ubqutous too o scc d grg. Th kds o pso to utz dpd o () th proprts o th uctos to b studd d (b) th proprts or chrctrstcs o th systm udr vstgto. Powr
More informationInternational Journal of Mathematical Archive-6(5), 2015, Available online through ISSN
Itratoal Joural of Mathmatal Arhv-6), 0, 07- Avalabl ol through wwwjmafo ISSN 9 06 ON THE LINE-CUT TRANSFORMATION RAPHS B BASAVANAOUD*, VEENA R DESAI Dartmt of Mathmats, Karatak Uvrsty, Dharwad - 80 003,
More information,. *â â > V>V. â ND * 828.
BL D,. *â â > V>V Z V L. XX. J N R â J N, 828. LL BL D, D NB R H â ND T. D LL, TR ND, L ND N. * 828. n r t d n 20 2 2 0 : 0 T http: hdl.h ndl.n t 202 dp. 0 02802 68 Th N : l nd r.. N > R, L X. Fn r f,
More information22 t b r 2, 20 h r, th xp t d bl n nd t fr th b rd r t t. f r r z r t l n l th h r t rl T l t n b rd n n l h d, nd n nh rd f pp t t f r n. H v v d n f
n r t d n 20 2 : 6 T P bl D n, l d t z d http:.h th tr t. r pd l 22 t b r 2, 20 h r, th xp t d bl n nd t fr th b rd r t t. f r r z r t l n l th h r t rl T l t n b rd n n l h d, nd n nh rd f pp t t f r
More informationCSE 373: More on graphs; DFS and BFS. Michael Lee Wednesday, Feb 14, 2018
CSE 373: Mor on grphs; DFS n BFS Mihl L Wnsy, F 14, 2018 1 Wrmup Wrmup: Disuss with your nighor: Rmin your nighor: wht is simpl grph? Suppos w hv simpl, irt grph with x nos. Wht is th mximum numr of gs
More information5/1/2018. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees
/1/018 W usully no strns y ssnn -lnt os to ll rtrs n t lpt (or mpl, 8-t on n ASCII). Howvr, rnt rtrs our wt rnt rquns, w n sv mmory n ru trnsmttl tm y usn vrl-lnt non. T s to ssn sortr os to rtrs tt our
More informationPH427/PH527: Periodic systems Spring Overview of the PH427 website (syllabus, assignments etc.) 2. Coupled oscillations.
Dy : Mondy 5 inuts. Ovrviw of th PH47 wsit (syllus, ssignnts tc.). Coupld oscilltions W gin with sss coupld y Hook's Lw springs nd find th possil longitudinl) otion of such syst. W ll xtnd this to finit
More informationINF5820 MT 26 OCT 2012
INF582 MT 26 OCT 22 H22 Jn Tor Lønnng l@.uo.no Tody Ssl hn rnslon: Th nosy hnnl odl Word-bsd IBM odl Trnng SMT xpl En o lgd n r d bygg..9 h.6 d.3.9 rgh.9 wh.4 buldng.45 oo.3 rd.25 srgh.7 by.3 onsruon.33
More informationP a g e 3 6 of R e p o r t P B 4 / 0 9
P a g e 3 6 of R e p o r t P B 4 / 0 9 p r o t e c t h um a n h e a l t h a n d p r o p e r t y fr om t h e d a n g e rs i n h e r e n t i n m i n i n g o p e r a t i o n s s u c h a s a q u a r r y. J
More informationSpin Structure of Nuclei and Neutrino Nucleus Reactions Toshio Suzuki
Spi Structur of Nucli d Nutrio Nuclus Rctios Toshio Suzuki Excittio of Spi Mods by s. Spctr DAR, DIF 3. Chrg-Exchg Rctios C, - N by iprovd spi-isospi itrctio with shll volutio Sprdig ffcts of GT strgth
More informationIFYFM002 Further Maths Appendix C Formula Booklet
Ittol Foudto Y (IFY) IFYFM00 Futh Mths Appd C Fomul Booklt Rltd Documts: IFY Futh Mthmtcs Syllbus 07/8 Cotts Mthmtcs Fomul L Equtos d Mtcs... Qudtc Equtos d Rmd Thom... Boml Epsos, Squcs d Ss... Idcs,
More informationTh pr nt n f r n th f ft nth nt r b R b rt Pr t r. Pr t r, R b rt, b. 868. xf rd : Pr nt d f r th B bl r ph l t t th xf rd n v r t Pr, 00. http://hdl.handle.net/2027/nyp.33433006349173 P bl D n n th n
More informationThe Z-Transform in DSP Lecture Andreas Spanias
The Z-Trsform DSP eture - Adres Ss ss@su.edu 6 Coyrght 6 Adres Ss -- Poles d Zeros of I geerl the trsfer futo s rtol; t hs umertor d deomtor olyoml. The roots of the umertor d deomtor olyomls re lled the
More informationIIT JEE MATHS MATRICES AND DETERMINANTS
IIT JEE MTHS MTRICES ND DETERMINNTS THIRUMURUGN.K PGT Mths IIT Trir 978757 Pg. Lt = 5, th () =, = () = -, = () =, = - (d) = -, = -. Lt sw smmtri mtri of odd th quls () () () - (d) o of ths. Th vlu of th
More informationA NEW GENERALIZATION OF KUMARASWAMY LINDLEY DISTRIBUTION
ou of Sttt: dv Thoy d ppto Vou 4 Nu 5 Pg 69-5 v t http://tfdv.o. DOI: http://d.do.og/.864/t_754 NW GNRLIZTION OF KUMRSWMY LINDLY DISTRIBUTION M. MHMOUD M. M. NSSR d M.. F Dptt of Mtht Futy of S Sh Uvty
More information